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Numerical relativistic hydrodynamics with ADER-WENO
adaptive mesh re�nement
Olindo Zanotti
University of Trento, Italy
6th East-Asian Numerical Astrophysics Meeting
15�19 September 2014 - Suwon
with: M. Dumbser
Outline
1 Motivations
2 The numerical method (ADER approach)
Finite volume scheme
Local space-time Discontinuous Galerkin predictor
3 Adaptive Mesh Re�nement
AMR implementation
Local Timestepping
4 The Richtmyer�Meshkov instability
1. Motivations
Motivations
Many physical problems show great disparities in the spatial and temporal
scales (multiscale problems), which a static grid approach cannot treat
e�ciently.
Turbulence
Magnetic instabilities
Magnetic reconnection
Binary mergers
Circumbinary discs
Relativistic jets
...
Adaptive Mesh Re�nement (AMR) , name-ly the possibility to change the computatio-nal grid dynamically in space and in time,becomes necessary.High Order Methods can also help substan-tially when very small details need to besolved.
Olindo Zanotti ADER-WENO Schemes with AMR 1 / 20
1. Motivations
Taken separately, AMR and High Order methods have long been used in RHD
AMR+High Order Methods (relativistic case)- FLASH code by Fryxell et al. (2000), ApJ Suppl., 131, 273- Zhang & MacFadyen (2006), ApJ Suppl., 164, 255- RENZO code by Wang et al. (2008), ApJ, 176, 467- PLUTO code by Mignone et al. (2012), ApJ Suppl., 198, 7
A particularly appealing implementation of High order methods is represented byADER schemes, which are one-step time-update schemes.- Original version: use the Lax-Wendro� procedure [Toro et al. (2001); Titarevand Toro (2002) ...]- Modern version: use a weak integral formulation of the governing PDE [Dumbseret al. (2008) JCP, 227, 3971; Balsara et al (2009) JCP, 228, 2480]
AMR+ADER schemes.... presented very recently for the non-relativistic Eulerequations by- Dumbser, Zanotti, Hidalgo, Balsara (2013), JCP, 248, 257-286- and extended to the relativistic regime at this conference for the �rst time
Olindo Zanotti ADER-WENO Schemes with AMR 2 / 20
1. Motivations
Taken separately, AMR and High Order methods have long been used in RHD
AMR+High Order Methods (relativistic case)- FLASH code by Fryxell et al. (2000), ApJ Suppl., 131, 273- Zhang & MacFadyen (2006), ApJ Suppl., 164, 255- RENZO code by Wang et al. (2008), ApJ, 176, 467- PLUTO code by Mignone et al. (2012), ApJ Suppl., 198, 7
A particularly appealing implementation of High order methods is represented byADER schemes, which are one-step time-update schemes.- Original version: use the Lax-Wendro� procedure [Toro et al. (2001); Titarevand Toro (2002) ...]- Modern version: use a weak integral formulation of the governing PDE [Dumbseret al. (2008) JCP, 227, 3971; Balsara et al (2009) JCP, 228, 2480]
AMR+ADER schemes.... presented very recently for the non-relativistic Eulerequations by- Dumbser, Zanotti, Hidalgo, Balsara (2013), JCP, 248, 257-286- and extended to the relativistic regime at this conference for the �rst time
Olindo Zanotti ADER-WENO Schemes with AMR 2 / 20
1. Motivations
Taken separately, AMR and High Order methods have long been used in RHD
AMR+High Order Methods (relativistic case)- FLASH code by Fryxell et al. (2000), ApJ Suppl., 131, 273- Zhang & MacFadyen (2006), ApJ Suppl., 164, 255- RENZO code by Wang et al. (2008), ApJ, 176, 467- PLUTO code by Mignone et al. (2012), ApJ Suppl., 198, 7
A particularly appealing implementation of High order methods is represented byADER schemes, which are one-step time-update schemes.- Original version: use the Lax-Wendro� procedure [Toro et al. (2001); Titarevand Toro (2002) ...]- Modern version: use a weak integral formulation of the governing PDE [Dumbseret al. (2008) JCP, 227, 3971; Balsara et al (2009) JCP, 228, 2480]
AMR+ADER schemes.... presented very recently for the non-relativistic Eulerequations by- Dumbser, Zanotti, Hidalgo, Balsara (2013), JCP, 248, 257-286- and extended to the relativistic regime at this conference for the �rst time
Olindo Zanotti ADER-WENO Schemes with AMR 2 / 20
1. Motivations
Taken separately, AMR and High Order methods have long been used in RHD
AMR+High Order Methods (relativistic case)- FLASH code by Fryxell et al. (2000), ApJ Suppl., 131, 273- Zhang & MacFadyen (2006), ApJ Suppl., 164, 255- RENZO code by Wang et al. (2008), ApJ, 176, 467- PLUTO code by Mignone et al. (2012), ApJ Suppl., 198, 7
A particularly appealing implementation of High order methods is represented byADER schemes, which are one-step time-update schemes.- Original version: use the Lax-Wendro� procedure [Toro et al. (2001); Titarevand Toro (2002) ...]- Modern version: use a weak integral formulation of the governing PDE [Dumbseret al. (2008) JCP, 227, 3971; Balsara et al (2009) JCP, 228, 2480]
AMR+ADER schemes.... presented very recently for the non-relativistic Eulerequations by- Dumbser, Zanotti, Hidalgo, Balsara (2013), JCP, 248, 257-286- and extended to the relativistic regime at this conference for the �rst time
Olindo Zanotti ADER-WENO Schemes with AMR 2 / 20
2. The numerical method (ADER approach) Finite volume scheme
Outline
1 Motivations
2 The numerical method (ADER approach)
Finite volume scheme
Local space-time Discontinuous Galerkin predictor
3 Adaptive Mesh Re�nement
AMR implementation
Local Timestepping
4 The Richtmyer�Meshkov instability
Olindo Zanotti ADER-WENO Schemes with AMR 2 / 20
2. The numerical method (ADER approach) Finite volume scheme
Finite volume scheme
We consider hyperbolic systems of balance laws in Cartesian coordinates
∂u
∂t+∂f
∂x+∂g
∂y+∂h
∂z= 0
We use a Standard Finite Volume discretization
un+1
ijk = unijk −
∆t
∆xi
(fi+ 1
2,jk − fi− 1
2,jk
)− ∆t
∆yj
(gi,j+ 1
2,k − gi,j− 1
2,k
)− ∆t
∆zk
(hij,k+ 1
2
− hij,k− 1
2
)+ ∆tSijk ,
over the control volumes Iijk = [xi− 1
2
; xi+ 1
2
]× [yj− 1
2
; yj+ 1
2
]× [zk− 1
2
; zk+ 1
2
], with
unijk =
1
∆xi
1
∆yj
1
∆zk
xi+ 1
2∫xi− 1
2
yj+ 1
2∫yj− 1
2
zk+ 1
2∫zk− 1
2
u(x , y , z , tn)dz dy dx
fi+ 1
2,jk =
1
∆t
1
∆yj
1
∆zk
tn+1∫tn
yj+ 1
2∫yj− 1
2
zk+ 1
2∫zk− 1
2
~f(q−h (xi+ 1
2
, y , z , t), q+h (xi+ 1
2
, y , z , t))dz dy dt, .
Olindo Zanotti ADER-WENO Schemes with AMR 3 / 20
2. The numerical method (ADER approach) Finite volume scheme
- Reconstruction: introduce a nodal basis of polynomials ψl (ξ) of order M de�ned withrespect to a set of Gauss-Legendre nodal points λk , such that ψl (λk) = δlk
Olindo Zanotti ADER-WENO Schemes with AMR 4 / 20
2. The numerical method (ADER approach) Finite volume scheme
- Build one�dimensional reconstruction stencils along each direction
Ss,xijk =i+R⋃
e=i−L
Iejk , Ss,yijk =
j+R⋃e=j−L
Iiek , Ss,zijk =k+R⋃
e=k−L
Iije ,
three stencils for even M
four stencils for odd M
M + 1 cells in each stencil
Olindo Zanotti ADER-WENO Schemes with AMR 5 / 20
2. The numerical method (ADER approach) Finite volume scheme
1 Perform an entire polynomial WENO reconstuction along x− direction:
ws,xh (x , tn) =
M∑p=0
ψp(ξ)wn,sijk,p := ψp(ξ)wn,s
ijk,p ∀ Ss,xijk
Impose integral conservation on all elements of the stencil:
1
∆xe
∫ xe+ 1
2
xe− 1
2
ψp(ξ(x))wn,sijk,p dx = u
nejk , ∀Iejk ∈ Ss,xijk
Perform a data-dependent nonlinear combination:
wxh(x , tn) = ψp(ξ)wn
ijk,p, with wnijk,p =
Ns∑s=1
ωswn,sijk,p
ωs =ωs∑k ωk
, ωs =λs
(σs + ε)r
σs = Σpmwn,sijk,pw
n,sijk,m Σpm =
M∑α=1
1∫0
∂αψp(ξ)
∂ξα· ∂
αψm(ξ)
∂ξαdξ .
[see Dumbser & Käser, (2007), JCP, 221, 693]
Olindo Zanotti ADER-WENO Schemes with AMR 6 / 20
2. The numerical method (ADER approach) Finite volume scheme
2 Perform a second polynomial WENO reconstuction along y− direction using asinput the M + 1 degrees of freedom wn
ijk,p:
ws,yh (x , y , tn) = ψp(ξ)ψq(η)wn,s
ijk,pq .
Apply integral conservation is now in the y direction:
1
∆ye
∫ ye+ 1
2
ye− 1
2
ψq(η(y))wn,sijk,pq dy = w
niek,p, ∀Iiek ∈ Ss,yijk .
=⇒
wy
h(x , y , tn) = ψp(ξ)ψq(η)wnijk,pq, with w
nijk,pq =
Ns∑s=1
ωswn,sijk,pq,
Olindo Zanotti ADER-WENO Schemes with AMR 7 / 20
2. The numerical method (ADER approach) Finite volume scheme
3 Perform the last polynomialWENO reconstuction along z− direction using as inputthe (M + 1)2 degrees of freedom wn
ijk,pq: We therefore have
ws,zh (x , y , z , tn) = ψp(ξ)ψq(η)ψr (ζ)wn,s
ijk,pqr .
with the integral conservation written as above,
1
∆ze
∫ ze+ 1
2
ze− 1
2
ψr (ζ(z))wn,sijk,pqr dz = w
niek,pq, ∀Iije ∈ Ss,zijk .
=⇒ The �nal three-dimensional WENO polynomial
wh(x, tn) = ψp(ξ)ψq(η)ψr (ζ)wnijk,pqr , w
nijk,pqr =
Ns∑s=1
ωswn,sijk,pqr .
Olindo Zanotti ADER-WENO Schemes with AMR 8 / 20
2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor
Outline
1 Motivations
2 The numerical method (ADER approach)
Finite volume scheme
Local space-time Discontinuous Galerkin predictor
3 Adaptive Mesh Re�nement
AMR implementation
Local Timestepping
4 The Richtmyer�Meshkov instability
Olindo Zanotti ADER-WENO Schemes with AMR 8 / 20
2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor
Local space-time DG predictor [see Dumbser et al. (2008), JCP, 227, 3971]
An alternative to the Cauchy-Kovalewski procedure to obtain the time evolution of thereconstructed polynomials and build one-step time-update numerical schemes.In practice we need
qh =⇒ fi+ 1
2,jk =
1
∆t
1
∆yj
1
∆zk
tn+1∫tn
yj+ 1
2∫yj− 1
2
zk+ 1
2∫zk− 1
2
~f(q−h (xi+ 1
2
, y , z , t), q+h (xi+ 1
2
, y , z , t))dz dy , dt,
∂u
∂τ+∂f∗
∂ξ+∂g∗
∂η+∂h∗
∂ζ= S
∗
with
f∗ =
∆t
∆xif, g
∗ =∆t
∆yjg, h
∗ =∆t
∆zkh, S
∗ = ∆tS.
We then multiply by the test function θp(ξ, τ) and integrate in space-time
1∫0
1∫0
1∫0
1∫0
θq
(∂u
∂τ+∂f∗
∂ξ+∂g∗
∂η+∂h∗
∂ζ− S
∗)dξdηdζdτ = 0.
where we choose a tensor product of the basis functionsθp(ξ, τ) = ψp(ξ)ψq(η)ψr (ζ)ψs(τ).
Olindo Zanotti ADER-WENO Schemes with AMR 9 / 20
2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor
Integration by parts in time yields...
1∫0
1∫0
1∫0
θq(ξ, 1)u(ξ, 1)dξdηdζ −1∫
0
1∫0
1∫0
1∫0
(∂
∂τθq
)udξdηdζdτ
+
1∫0
1∫0
1∫0
1∫0
[θq
(∂f∗
∂ξ+∂g∗
∂η+∂h∗
∂ζ− S
∗)]
dξdηdζdτ
=
1∫0
1∫0
1∫0
θq(ξ, 0)wh(ξ, tn)dξdηdζ.
We introduce the discrete space-time solution qh
qh = qh(ξ, τ) = θp (ξ, τ) qp,
and similarly for the �uxes and sources
f∗h = θp f
∗p = θpf
∗ (qp) , . . . S∗h = θpS
∗p = θpS
∗ (qp) .
Insert everything in Eq. above and obtain....
Olindo Zanotti ADER-WENO Schemes with AMR 10 / 20
2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor
1∫0
1∫0
1∫0
θq(ξ, 1)θp(ξ, 1)qp dξdηdζ −1∫
0
1∫0
1∫0
1∫0
(∂
∂τθq
)θpqp dξdηdζdτ
+
1∫0
1∫0
1∫0
1∫0
[θq
(∂
∂ξθp f
∗p +
∂
∂ηθpg
∗p +
∂
∂ζθph
∗p − θpS∗
p
)]dξdηdζdτ
=
1∫0
1∫0
1∫0
θq(ξ, 0)wh(ξ, tn) dξdηdζ ,
which is a nonlinear algebraic equation system for the unknown coe�cients qp.In a more compact form:
K1qpqp + K
ξqp · f∗p + K
ηqpg
∗p + K
ζqph
∗p = MqpS
∗p + F
0qmw
nm,
with the various matrices de�ned as
K1
qp =
1∫0
1∫0
1∫0
θq(ξ, 1)θp(ξ, 1)dξ −1∫0
1∫0
1∫0
1∫0
(∂
∂τθq
)θpdξdτ,
Kξqp =
(Kξqp,K
ηqp,K
ζqp
)=
1∫0
1∫0
1∫0
1∫0
θq∂
∂ξθpdξdτ,
Mqp =
1∫0
1∫0
1∫0
1∫0
θqθpdξdτ, F0
qp =
1∫0
1∫0
1∫0
θq(ξ, 0)ψm(ξ)dξ,
Olindo Zanotti ADER-WENO Schemes with AMR 11 / 20
2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor
The product of the matrices with the vectors of degrees of freedom can be e�ciently
implemented in a dimension�by�dimension manner. An iterative scheme can be adopted
[see Dumbser & Zanotti (2009), JCP, 228, 6991]
K1qpq
k+1p −MqpS
∗,k+1p = F0
qmwnm −K
ξqp · f
∗,kp −K
ηqpg∗,kp −K
ζqph∗,kp
Once this is done, we have everything to write the scheme as
un+1ijk = un
ijk −∆t
∆xi
(fi+ 1
2,jk − fi− 1
2,jk
)− ∆t
∆yj
(gi ,j+ 1
2,k − gi ,j− 1
2,k
)− ∆t
∆zk
(hij ,k+ 1
2− hij ,k− 1
2
)+ ∆tSijk ,
Local Lax-Friedrichs �ux
~f(q−h , q
+h
)=
1
2
(f(q−h ) + f(q+h )
)− 1
2|smax|
(q+h − q
−h
), (1)
where |smax| denotes the maximum absolute value of the eigenvalues of theJacobian matrix A = ∂f/∂u.
Osher�type �ux
~f(q−h , q
+h
)=
1
2
(f(q−h ) + f(q+h )
)− 1
2
(∫ 1
0
|A(ψ(s))| ds)(
q+h − q
−h
), (2)
Olindo Zanotti ADER-WENO Schemes with AMR 12 / 20
3. Adaptive Mesh Re�nement AMR implementation
Outline
1 Motivations
2 The numerical method (ADER approach)
Finite volume scheme
Local space-time Discontinuous Galerkin predictor
3 Adaptive Mesh Re�nement
AMR implementation
Local Timestepping
4 The Richtmyer�Meshkov instability
Olindo Zanotti ADER-WENO Schemes with AMR 12 / 20
3. Adaptive Mesh Re�nement AMR implementation
AMR implementation
We have developed a cell-by-cell AMR technique in which the computational
domain is discretized with a uniform Cartesian grid at the coarsest level.
Adopt a re�nement criterion, marking a cell for re�nement if
χm > χref , where
χm =
√ ∑k,l (∂
2Φ/∂xk∂xl )2∑k,l [(|∂Φ/∂xk |i+1 + |∂Φ/∂xk |i )/∆xl + ε|(∂2/∂xk∂xl )||Φ|]2
.
When a cell of the level ` is re�ned, it is subdivided as
∆x` = r∆x`+1 ∆y` = r∆y`+1 ∆z` = r∆z`+1 ∆t` = r∆t`+1
Each cell Cm, at any level of re�nement, has one among three possible
status �ags.
- active cell
- virtual child cell
- virtual mother cellOlindo Zanotti ADER-WENO Schemes with AMR 13 / 20
3. Adaptive Mesh Re�nement AMR implementation
...AMR implementation
If the Voronoi neighbors of an active re�ned cell Cm are not themselves at thesame level of re�nement of Cm, they have virtual children at the same level ofre�nement of Cm.In order to keep the reconstruction local on the coarser grid level, we have r ≥ M.
The levels of re�nement of two cells that are Voronoi neighbors of each other canonly di�er by at most unity.
Olindo Zanotti ADER-WENO Schemes with AMR 14 / 20
3. Adaptive Mesh Re�nement AMR implementation
The beauty of the local-spacetime DG predictor: it does not need
any exchange of information with neighbor elements, even if two
adjacent cells are on di�erent levels of re�nement.
Projection
Projection is the typical AMR operation, by which an active mother
assigns values to the virtual children (σ = 1) at intermediate times
um(tn` ) =1
∆x`
1
∆y`
1
∆z`
∫Cm
qh(x, tn` )dx. (3)
Needed for performing the reconstruction on the �ner grid level at
intermediate times.
Averaging
Averaging is another typical AMR operation by which a virtual mother
cell (σ = −1) obtains its cell average by averaging recursively over the
cell averages of all its children at higher re�nement levels.
um =1
rd
∑Ck∈Bm
uk . (4)
Olindo Zanotti ADER-WENO Schemes with AMR 15 / 20
3. Adaptive Mesh Re�nement Local Timestepping
Outline
1 Motivations
2 The numerical method (ADER approach)
Finite volume scheme
Local space-time Discontinuous Galerkin predictor
3 Adaptive Mesh Re�nement
AMR implementation
Local Timestepping
4 The Richtmyer�Meshkov instability
Olindo Zanotti ADER-WENO Schemes with AMR 15 / 20
3. Adaptive Mesh Re�nement Local Timestepping
Local Timestepping
Every re�nement level is advanced in time with its local timestep
∆t` = r∆t`+1. Update criterion:
tn+1` ≤ tn+1
`−1 , 0 ≤ ` ≤ `max, (5)
Starting from the common initial time t = 0, the �nest level of re�nement
`max is evolved �rst and performs a number of r sub-timesteps before the
next coarser level `max − 1 performs its �rst time update.
=⇒ A total amount of r` sub-timesteps on each level are performed in
order to reach the time tn+10 of the coarsest level.
Olindo Zanotti ADER-WENO Schemes with AMR 16 / 20
4. The Richtmyer�Meshkov instability
RHD equations
∂tu + ∂i fi = 0 ,
The conservative variables and the corresponding �uxes in the i direction
are
u =
D
Sj
E
, f i =
v iD
W ij
S i
. (6)
D = ργ,
Si = ρhγ2vi ,
E = ρhγ2 − p,
where γ = (1− v2)−1/2 is the Lorentz factor of the �uid with respect to
the laboratory observer and
Wij ≡ ρhγ2vivj + pδij
Olindo Zanotti ADER-WENO Schemes with AMR 17 / 20
4. The Richtmyer�Meshkov instability
Just one test...
Figura : Standard Riemann problem: the shock wave is resolved within one single
cell!!
Olindo Zanotti ADER-WENO Schemes with AMR 18 / 20
4. The Richtmyer�Meshkov instability
The Richtmyer�Meshkov instability
The Richtmyer�Meshkov (RM) instability is a typical �uid instability which
develops when a shock wave crosses a contact discontinuity within a �uid,
or between two di�erent �uids.
Relevant in
Inertial Con�nement Fusion
Supernovae remnant formation
Relativistic jets
Figura : Schematic representation of the initial conditions in a representative
model with Atwood number A > 0.
Olindo Zanotti ADER-WENO Schemes with AMR 19 / 20
Conclusions
We have presented the �rst ADER-WENO �nite volume scheme for relativistichydrodynamics on AMR grids
Compared to Runge�Kutta time stepping, the use of a high order one�step schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.
High Order in combination with AMR is extremely bene�cial, in particular whensmall scale structures need to be solved
When applied to the study of the Richtmyer�Meshkov instability, new interestingresults are obtained
Conclusions
We have presented the �rst ADER-WENO �nite volume scheme for relativistichydrodynamics on AMR grids
Compared to Runge�Kutta time stepping, the use of a high order one�step schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.
High Order in combination with AMR is extremely bene�cial, in particular whensmall scale structures need to be solved
When applied to the study of the Richtmyer�Meshkov instability, new interestingresults are obtained
Conclusions
We have presented the �rst ADER-WENO �nite volume scheme for relativistichydrodynamics on AMR grids
Compared to Runge�Kutta time stepping, the use of a high order one�step schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.
High Order in combination with AMR is extremely bene�cial, in particular whensmall scale structures need to be solved
When applied to the study of the Richtmyer�Meshkov instability, new interestingresults are obtained
Conclusions
We have presented the �rst ADER-WENO �nite volume scheme for relativistichydrodynamics on AMR grids
Compared to Runge�Kutta time stepping, the use of a high order one�step schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.
High Order in combination with AMR is extremely bene�cial, in particular whensmall scale structures need to be solved
When applied to the study of the Richtmyer�Meshkov instability, new interestingresults are obtained
Conclusions
We have presented the �rst ADER-WENO �nite volume scheme for relativistichydrodynamics on AMR grids
Compared to Runge�Kutta time stepping, the use of a high order one�step schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.
High Order in combination with AMR is extremely bene�cial, in particular whensmall scale structures need to be solved
When applied to the study of the Richtmyer�Meshkov instability, new interestingresults are obtained